I'm looking for an example of a differentiable manifold of class $C^k$ but not class $C^{k +1}.$ I found an exercise in Hirsh's book, which suggests that the graph of $f (x) = |x|^{\lambda}$, where $k<\lambda<k+1$ is a differentiable manifold of class $C^k$ but not class $C^{k+1}$. I initially thought the atlas given by the graph coordinates , but obviously does not work.
My problem:what kind of differentiable structure I need for the graph of $ f $ is a differentiable manifold of class $C^k$ but not $C^{k +1}$